\section{TSV RLCG model in AC and high frequency region}\label{sec:level2}
Previous section introduces papers that used simple RLC model for DC and low frequency region. In these models, the resistance and inductance are in serial and capacitance exists between TSV and the substrate. But these papers treated silicon substrate as an ideal conductor, the dielectric layer and depletion region are neglected or simplified, causing higher error compared to realistic cases. 

AC conduction in silicon, skin effect and eddy currents in silicon should be also taken into consideration for TSV modeling at AC and high frequencies \cite{Xu2010}. Skin effect means the current density drops by a certain factor below the surface of a conductor. It has great impact on the high frequency resistance. RLCG model is usually used for high frequency TSV modeling \cite{Xu2010, Ryu2005}. The model contains two components: admittance per unit TSV height which consists of conductance and susceptance; impedance per unit TSV height which is composed of resistance and reactance. 

In this section, we are mainly focus on the RLCG model developed in paper \cite{Xu2010}. The equivalent distributed circuit model is shown in figure \ref{fig:RLCG}(a), the simplified model is given in figure \ref{fig:RLCG}(b). The impedance which is represented by $Z$ is inside TSV, similar to the resistance and inductance in serial in RLC model. Capacitance exists inside the silicon depletion region. Admittance, represented by $Y$ exists in the silicon substrate between two adjacent TSVs. In this figure, $Y_{open}$ is the input admittance between ports 1 and 2 if ports 3 and 4 are open while $Z_{short}$ represents the impedance between ports 1 and 2 when ports 3 and 4 are short circuited.

\begin{figure}
\includegraphics[width=0.40\textwidth]{figures/RLCG.pdf}
\caption{RLCG model for TSV, (a) the equivalent distributed RLCG model of two coupled TSVs; (b) a simplified distributed transmission line model. }\cite{Xu2010}\label{fig:RLCG}
\end{figure}

\subsection{Admittance of TSV in RLCG model}
The admittance (CG) per unit TSV height can be treated as two components working in serial, one is the silicon depletion region ($C_1$) and the other is the coupling admittance ($Y_2$) in the bulk silicon. The admittance formula is shown in the following equation:
\begin{equation}
Y=[2(j\omega C_1)^{-1} + Y_2^{-1}]^{-1}
\end{equation}
where $\omega$ is the radial frequency. Since there are two TSVs contribute to $C_1$ in serial with $Y_2$, the equation contains a factor of 2 before $C_1$. 

The capacitance in the depletion region ($C_1$) can be captured with the following formula:
\begin{equation}
C_1=[ \frac{1}{2\pi \varepsilon_{OX}} \cdot ln \left( 1+\frac{t_{OX}}{r_{via}} \right)+
\frac{1}{2\pi \varepsilon_{Si}}\cdot ln\left(1+\frac{\omega_{dep}}{r_{via}+t_{OX}}\right)]^{-1}
\end{equation}
where $\varepsilon_{OX}$ and $\varepsilon_{Si}$ represent the permittivity of dielectric and silicon substrate, respectively, the geometrical parameters $r_{via}$, $t_{OX}$, and $\omega_{dep}$ are the via radius, isolation layer thickness and silicon depletion width. 

The coupling admittance in Si bulk can be illustrated with following equation:
\begin{equation}
Y_2=\pi(\sigma_{Si}+j\omega \varepsilon_{Si})/arccosh\left(\frac{d}{2}/(r_{via}+t_{OX}+\omega_{dep}) \right )
\end{equation}
where $\sigma_{Si}$ is the conductivity of silicon. 

The CG model is verified with a 2-D quasi-electrostatic simulation tool. The results suggest that at low frequencies, if the depletion region is not considered, the error is not negligible, however this difference is not so significant at high frequencies. 

\subsection{Impedance of TSV in RLCG model}
The serial impedance (RL) per unit height is not so straight forward. The final expression results are shown here without detail deduction steps. For simplicity, the serial impedance can be treated as the sum of three components: the inner impedance of TSV ($Z_{metal}$), the outer inductance ($L_{outer}$), and the resistance due to eddy currents in silicon substrate ($R_{sub}$):
\begin{equation}
Z = 2Z_{metal} + j\omega L_{outer} + R_{sub}
\end{equation}

The equations for these three components are given as followings:
\begin{equation}
Z_{metal}=\frac{(1-j)\cdot J_0((1-j)r_{via}/\delta _{metal})}{\sigma _{metal} \cdot 2\pi r_{via}\delta_{metal} \cdot J_1((1-j)r_{via}/\delta_{metal})}
\end{equation}

\begin{eqnarray}
\lefteqn{R_{sub}\approx \frac{\omega \mu}{2} \cdot} \\
& Re\left[H_0^{(2)}\left(\frac{1-j}{\delta_{Si}}9r_{via}+t_{OX}+\omega_{dep}-H_0^{(2)}\left(\frac{(1-j)d}{\delta_{Si}} \right ) \right ) \right ] \nonumber
\end{eqnarray}

\begin{eqnarray}
\lefteqn{L_{outer} \approx \frac{\mu}{\pi}ln\left(\frac{r_{via}+t_{OX}+\omega_{dep}}{r_{via}}\right) +\frac{\mu}{2} \cdot  } \\
& Im\left[ H_0^{(2)}\left( \frac{1-j}{\delta_{Si}}(r_{via}+t_{OX}+\omega_{dep}\right) -H_0^{(2)}\left(\frac{(1-j)d}{\delta_{Si}}\right)\right] \nonumber
\end{eqnarray}
where $\mu$ is the permeability in either TSV or silicon, $J_0$ is the 0th order Bessel function of the first type, $H_0^{(2)}$ is the 0th order Hankel function of the second type, $\delta_{metal}$ and $\delta_{Si}$ are the damping parameters for TSV and silicon, respectively.

The results are compared with simulation tool and indicate that taking into consideration the skin effect in TSV is of great importance for high frequency analysis when the higher frequency resistance is dominant over DC resistance.

\subsection{TSV electrical performance with RLCG model}
As technology scales, the diameter and pitch of TSVs will reduce, however, the substrate thickness almost remains the same as predicted by the International Technology Roadmap for Semiconductors (ITRS). When the radius of TSVs reduce, C, G and L do not change much due to the proportional scaling of geometrical parameters. However, resistance increases significantly when the frequency reaches 1GH and 100GHz due to the decreasing TSV cross area. 

In terms of circuit performance sensitivity, capacitance has the most important impact on circuit behavior while resistance is of the least importance. The circuit exhibits the short-transmission line behavior on signal propagation, which indicates that simple RLC model is enough for delay and signal rise/fall calculation. However, the L and G are crucial factors for the estimation of voltage variations in $V_{DD}$ and $GND$. More accurate whole circuit performance evaluation can only be done with all RLCG components.